Optimal growth of entire functions frequently hypercyclic for the differentiation operator

We solve a problem posed by Bonilla and Grosse-Erdmann (2007) [7] by constructing an entire function f which is frequently hypercyclic with respect to the differentiation operator, and satisfies Mf(r)⩽cerr−1/4, where c>0 may be chosen arbitrarily small. This growth rate is sharp. We also obtain optimal results for minimal growth in terms of average Lp-norms. Among other tools, the proof uses the Rudin–Shapiro polynomials and heat kernel estimates.